Hamiltonicity, pancyclicity, and full cycle extendability in multipartite tournaments

Abstract

Abstract A digraph with vertices is Hamiltonian (pancyclic and vertex‐pancyclic, respectively) if contains a Hamilton cycle (a cycle of every length , for every vertex , a cycle of every length through , respectively.) It is well‐known that a strongly connected tournament is Hamiltonian, pancyclic, and vertex pancyclic. A digraph is cycle extendable if for every non‐Hamiltonian cycle of , there is a cycle such that contains all vertices of plus another vertex of . A cycle extendable digraph is fully cycle extendable if for every vertex , there exists a cycle of length 3 through . Note that full cycle extendability is a stronger property than vertex pancyclicity. While it is well‐known that every strongly connected tournament is vertex pancyclic, Hendry showed that not every strongly connected tournament is fully cycle extendable and characterized an infinite class of strongly connected tournaments, which are not fully cycle extendable. A ‐partite tournament is an orientation of a ‐partite complete graph (for , it is called a bipartite tournament). Gutin and later Häggkvist and Manoussakis characterized Hamiltonian bipartite tournaments. A bipartite digraph with vertices is even pancyclic (even vertex pancyclic, respectively) if contains a cycle of every even length (a cycle of every even length through for every , respectively). Beineke and Little, and Zhang proved that every bipartite tournament is even pancyclic and even vertex pancyclic, respectively, if and only if it is Hamiltonian and does not belong to a well‐defined infinite class of regular bipartite tournaments. We prove that unlike in the case of tournaments, every even pancyclic bipartite tournament is fully cycle extendable. We show that this result cannot be extended to ‐partite tournaments for any fixed (where we naturally replace even vertex pancyclicity by vertex pancyclicity).